Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
216 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7×AGL(1,𝔽5) |
| commutator, hypocenter | C5×C7 | C2×C42 |
| Frattini | C2 | C2×D7×AGL(1,𝔽5) |
| Fitting, socle | C22×C5×C7 | C2×C4 |
| prime radical | C22×C7×D5 | C22 |
| cosocle | C2×C7×D5 | C23 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. AGL(1,𝔽5)×Dic7 | C2 | direct (N × H) |
| 6. C2×C7⋊AGL(1,𝔽5) | C4 | semidirect |
| 8. C2×C7×AGL(1,𝔽5) | C4 | semidirect |
| 10. C2×Dic35 | C4 | semidirect |
| 11. C2×C5×Dic7 | C4 | semidirect |
| 12. D5×Dic7 | C4 | semidirect |
| 21. C7⋊AGL(1,𝔽5) | C2×C4 | semidirect |
| 22. C7×AGL(1,𝔽5) | C2×C4 | semidirect |
| 23. Dic35 | C2×C4 | semidirect |
| 24. C5×Dic7 | C2×C4 | semidirect |
| 33. C2×C5×C7 | C42 | semidirect |
| 35. C7×D5 | C42 | semidirect |
| 46. C2×Dic7 | AGL(1,𝔽5) | direct (N × H) |
| 50. C2×AGL(1,𝔽5) | Dic7 | direct (N × H) |
| 56. C5×C7 | C2×C42 | semidirect |
| 66. Dic7 | C2×AGL(1,𝔽5) | direct (N × H) |
| 73. AGL(1,𝔽5) | C2×Dic7 | direct (N × H) |
| 83. C2×C7 | C4×AGL(1,𝔽5) | semidirect |
| 85. C2×C5 | C4×Dic7 | semidirect |
| 87. D5 | C4×Dic7 | semidirect |
| 99. C7 | C2×C4×AGL(1,𝔽5) | semidirect |
| 100. C5 | C2×C4×Dic7 | semidirect |
| 113. C2 | AGL(1,𝔽5)×Dic7 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7⋊AGL(1,𝔽5) | 2 |
| C22×C7×AGL(1,𝔽5) | 2 |
| C2×D5×Dic7 | 2 |
| AGL(1,𝔽5)×Dic7 | 2 |
| C2×C4×Dic7 | 5 |
| C2×C4×AGL(1,𝔽5) | 7 |